Let $X$ be a poissonian random variable with an expected value of 100. (I guess it means $\lambda=100$). Let there be a kindergarten with $X$ kids. On a certain day, the teacher decides to split the kids to two groups- Heads group, and Tails group. For every kid she tosses a coin and that way knows what group the kid belongs to, independently. If $Z_H$ is the number of kids in the Heads group, what is its distribution?
a. Negative binomial
I am really confused about it. How does one determines such a distribution? How does the fact that $X$ is poissonian help here? If I am to look into $Z_H$ distribution, isn't $X$ to merely be taken as a number? I would really appreciate your help.